Question

Multiple Choice Questions 1. (2 points) Suppose you have a variable Y i ​ , and a dummy variable X i ​ . If we call p= P(X i ​ =1), find an equation for the CEF of Y i ​ with respect to X i ​ . A. m(x)=E[Y i ​ ∣X i ​ =1]p+E[Y i ​ ∣X i ​ =0](1−p) B. m(x)=E[Y i ​ ∣X i ​ =1]p C. m(x)=xE[Y i ​ ∣X i ​ =1]+(1−x)E[Y i ​ ∣X i ​ =0] D. m(x)=E[Y i ​ ∣X i ​ =x](1−p)+E[Y i ​ ∣X i ​ =x]p E. m(x)=pE[Y i ​ ∣X i ​ =x]x 2. (2 points) Continuing from Question 1, how could you estimate β 1 ​ in a regression of Y i ​ on X i ​ ? A. β ^ ​ 1 ​ = p ^ ​ B. β ^ ​ 1 ​ = n 1 ​ 1 ​ ∑ i∈S 1 ​ ​ Y i ​ − n 0 ​ 1 ​ ∑ i∈S 0 ​ ​ Y i ​ C. β ^ ​ 1 ​ =1− p ^ ​ D. β ^ ​ 1 ​ =m(1)−m(0) E. None of these

Answer 1

The Conditional Expectation Function (CEF) for
`\(Y_i\)` with respect to the dummy variable
`\(X_i\)` is represented as
`\(m(x) = xE[Y_i \mid X_i = 1] + (1 - x)E[Y_i \mid X_i = 0]\).` To estimate the slope
`(\(\beta_1\))` in a regression of
`\(Y_i\) on \(X_i\),` the formula is
`\(\hat{\beta}_1 = m(1) - m(0)\),`indicating the difference in conditional expectations between
`\(X_i = 1\) and \(X_i = 0\) in`the regression model.

Step-by-step explanation:

Certainly, let's break down the calculation and explanation for each part:

1. Conditional Expectation Function (CEF):

The CEF, denoted as
`\( m(x) \),` is defined as the expected value of
`\( Y_i \) given \( X_i = x \).`The correct formula is
`\( m(x) = xE[Y_i \mid X_i = 1] + (1 - x)E[Y_i \mid X_i = 0] \).`

• 
  `\( E[Y_i \mid X_i = 1] \): This is the expected value of \( Y_i \) when \( X_i = 1 \).`

• 
  `\( E[Y_i \mid X_i = 0] \):` This is the expected value of
  `\( Y_i \) when \( X_i = 0 \).`

`\( x \) and \( (1 - x) \):` These are weights that represent the proportion of the values of
`\( Y_i \)` associated with
`\( X_i = 1 \) and \( X_i = 0 \)` respectively.

Therefore, the formula
`\( m(x) = xE[Y_i \mid X_i = 1] + (1 - x)E[Y_i \mid X_i = 0] \)` is a weighted average of the expected values based on the values of
`\( X_i \).`

2. Estimating
`\( \beta_1 \)` in Regression:

To estimate
`\( \beta_1 \)` in a regression of
`\( Y_i \) on \( X_i \),` the slope can be obtained by taking the difference between the conditional expectations of
`\( Y_i \) given \( X_i = 1 \) and \( X_i = 0 \).` Therefore,
`\( \hat{\beta}_1 = m(1) - m(0) \).`

• 
  `\( m(1) \):`The conditional expectation of
  `\( Y_i \) when \( X_i = 1 \).`

• 
  `\( m(0) \):` The conditional expectation of
  `\( Y_i \) when \( X_i = 0 \).`

• 
  `\( \hat{\beta}_1 \):`The estimated slope in the regression of
  `\( Y_i \) on \( X_i \).`

This formula reflects the change in the expected value of
`\( Y_i \)` for a one-unit change in
`\( X_i \),` providing an estimate of the slope in the regression model.